In this section, we construct the family of graphs that we will study in the rest of this paper. Our construction is based on the construction in Section 3.2 of [1]}. The authors of [1]} introduced a process to construct an intrinsic graph \(\Gamma _\psi \)  that is far from a vertical plane at many scales (see Proposition 3.4 of [1]}). Unfortunately for our purposes, the intrinsic gradient \(\nabla _\psi \psi \)  is \(L_2\) –bounded but not bounded, so \(\Gamma _\phi \)  is not intrinsic Lipschitz. In this section, we will modify that construction via a stopping time argument so that it produces an intrinsic Lipschitz function with similar properties. To keep this paper self-contained, we will reproduce the construction of [1]} in parallel with our modification.
